On the exponent of a certain quotient of Whitehead groups of division algebras

Abstract

Let D be an F-central division algebra. In this paper, we investigated the exponent of the group G ( D ) = D * / Nrd D ( D * ) D ′ , where D * is the group of units of D, Nrd D ( D * ) is the image of D * under the reduced norm map and D ′ is the commutator subgroup of D * . We show that if   exp   ( G ( D ) ) < ind ( D ) , then D and F satisfy strong conditions. In particular, we observe that if D is a sum cyclic algebras in Br ( F ) , then   exp   ( G ( D ) ) < ind ( D ) if and only if F is euclidean and D is a tensor product of an ordinary quaternion algebra and a division algebra of odd index.